DOCSLIB.ORG

Chaos and the (Un)Predictability of Evolution in a Changing Environment

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chaos and the (Un)Predictability of Evolution in a Changing Environment bioRxiv preprint doi: https://doi.org/10.1101/222471; this version posted November 21, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Chaos and the (un)predictability of evolution in a changing environment Artur Rego-Costaa,c, Florence Débarreb, and Luis-Miguel Chevinc,1 aDepartment of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, USA bCentre Interdisciplinaire de Recherche en Biologie (CIRB), Collège de France, CNRS UMR 7241 - Inserm U1050, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France cCEFE-CNRS, UMR 5175, 1919 route de Mende, 34293 Montpellier Cedex 05, France 1To whom correspondence should be addressed. E-mail: [email protected] November 20, 2017 Among the factors that may reduce the predictability of evolution, chaos, characterized by a strong de- pendence on initial conditions, has received much less attention than randomness due to genetic drift or environmental stochasticity. It was recently shown that chaos in phenotypic evolution arises commonly un- der frequency-dependent selection caused by competitive interactions mediated by many traits. This result has been used to argue that chaos should often make evolutionary dynamics unpredictable. However, pop- ulations also evolve largely in response to external changing environments, and such environmental forcing is likely to influence the outcome of evolution in systems prone to chaos. We investigate how a changing environment causing oscillations of an optimal phenotype interacts with the internal dynamics of an eco- evolutionary system that would be chaotic in a constant environment. We show that strong environmental forcing can improve the predictability of evolution, by reducing the probability of chaos arising, and by damp- ening the magnitude of chaotic oscillations. In contrast, weak forcing can increase the probability of chaos, but it also causes evolutionary trajectories to track the environment more closely. Overall, our results indicate that, although chaos may occur in evolution, it does not necessarily undermine its predictability. Keywords Eco-evolutionary dynamics, Adaptation to changing environments, Predictability, Repeatability, Chaotic dynamics Introduction ics of a system, despite being completely predictable from the initial conditions, are strongly sensitive to The extent to which evolution is repeatable and pre- them. Under chaotic dynamics, any measurement er- dictable bears on the usefulness of evolutionary biol- ror, regardless how small, will be amplified over time, ogy as a tool for a growing number of applied fields, to the point that it becomes impossible to make ac- including drug resistance management in pests and curate predictions beyond a certain timescale (Ott, pathogens, or sustainable agriculture and harvesting 2002; Petchey et al., 2015). under climate change. So far, the investigation of fac- It was recently demonstrated theoretically that evo- tors that may reduce the predictability of evolution lutionary dynamics at the phenotypic level can be- has mostly focused on various sources of stochastic- come chaotic when natural selection results from ity (i.e. randomness), namely genetic drift, the con- between-individual interactions within a species tingency of mutations, and randomly fluctuating en- (Doebeli & Ispolatov, 2014), i.e., even in the ab- vironments (Crow & Kimura, 1970; Lenormand et al., sence of any interspecific eco-evolutionary feed- 2009; Sæther & Engen, 2015). A much less explored backs, which are known to enhance ecological chaos source of unpredictability in evolution is determin- (Abrams & Matsuda, 1997; Dercole & Rinaldi, 2010; istic chaos (but see Hamilton, 1980; Altenberg, 1991; Dercole et al., 2010). Specifically, evolutionary chaos Gavrilets & Hastings, 1995; Abrams & Matsuda, 1997; arises in single-species models when (i) selection is Dercole & Rinaldi, 2010; Dercole et al., 2010; Doebeli frequency-dependent, such that the fitness of an in- & Ispolatov, 2014), which occurs when the dynam- dividual depends on trait-mediated interactions with 1 bioRxiv preprint doi: https://doi.org/10.1101/222471; this version posted November 21, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. conspecifics; (ii) the fitness effects of these interac- ing phenotypic selection, modeled as a moving opti- tions are not simply a function of the phenotypic dif- mal phenotype (a classic approach, reviewed by Kopp ference between the interactors (unlike typical mod- & Matuszewski, 2014), influences the predictability els of within- and between-species interactions, e.g. of evolutionary dynamics in a context where chaos Dieckmann & Doebeli, 1999; Doebeli & Ispolatov, is expected to make evolution highly unpredictable 2010); and (iii) the number of traits involved in these in the absence of environmental forcing. Focusing interactions (described as organismal complexity) is on a periodic environment, we ask how the ampli- large (Doebeli & Ispolatov, 2014).The authors con- tude and period of cycles influence (i) the probability cluded from this study that evolution is likely to that evolutionary trajectories are chaotic, and (ii) the be much less predictable than generally perceived. degree to which those trajectories that are indeed However, the theoretical demonstration that chaos is chaotic track the optimal phenotype set by the en- possible in a system is not sufficient to argue that this vironment, making them partly predictable.We show system is necessarily unpredictable, as we elaborate that a changing environment can dramatically alter below. the probability of evolutionary chaos in either direc- First, the parameter values that lead to chaos may tion, but that evolutionary tracking of the environ- be rare in nature (Hastings et al., 1993; Zimmer, 1999). mental forcing generally contributes to making evo- For instance in ecology, chaos has long been known lutionary trajectories much more predictable than to be a possible outcome of even simple population anticipated from a theory that ignores any role of the dynamic models (May, 1976), but despite a few clear external environment. This suggests that the pre- empirical demonstrations in the laboratory (Benincà dictability of evolution is partly determined by a bal- et al., 2008) and in the wild (Benincà et al., 2015), ance between the strength of intraspecific interac- most natural populations seem to have demographic tions and responses to environmental change. parameters placing them below the “edge of chaos” (Hastings et al., 1993; Ellner & Turchin, 1995; Zimmer, 1999; Dercole et al., 2006). Methods Second, and importantly with respect to evolution, Model a potentially chaotic system may still be predictable because it is subject to forcing by external factors with We consider a set of d phenotypic traits that evolve autonomous dynamics. In eco-evolutionary pro- under both frequency-dependent and frequency- cesses, such external forcing generally results from independent selection. Frequency-independent se- a changing environment that affects fitness compo- lection is assumed to be caused by stabilizing selec- nents and their dependence on phenotypes, causing tion towards an optimal trait value, at which carry- variation in natural selection. In fact, environmental ing capacity K is maximized. For instance, selection variability affecting population growth (Lande et al., on beak size/shape in a bird, or mouth shape in a 2003; Ellner et al., 2011; Pelletier et al., 2012) and nat- fish, may have a local optimum set by the available ural selection (MacColl, 2011; Chevin et al., 2015) is type of resources (Martin & Wainwright, 2013; Grant probably ubiquitous in natural populations, as doc- & Grant, 2014). The frequency-dependent compo- umented notably by numerous examples of ecologi- nent of selection, on the other hand, emerges from cal and evolutionary responses to climate change (re- trait-mediated ecological and social interactions be- viewed by Davis et al., 2005; Parmesan, 2006; Hoff- tween individuals within the species (either coopera- mann & Sgrò, 2011). When such external forcing is tive or competitive). Selection on a bird’s beak mor- imposed, the predictability of evolution is likely to phology, for instance, depends not only on the avail- change, because (i) forcing can alter the probability able types of resources, but also on competition with that the system is indeed chaotic, for instance by sup- conspecifics for these resources (as in Grant & Grant, pression of chaos through synchronization to the ex- 2014). The intensity of this competition may depend ternal forcing (e.g. Pikovsky et al., 2003); and, (ii) even on the beak size of competing individuals, but also if the dynamics remain chaotic, they may still be af- on other traits of these interactors, such as their ag- fected by the forcing in ways that make them largely gressiveness, territoriality, or degree of choosiness in predictable. food preference. This frequency-dependent compo- We investigate how a changing environment affect- nent of selection, when it involves many traits, can 2 bioRxiv preprint doi: https://doi.org/10.1101/222471; this version posted November 21,
Load more

Recommended publications
  • A General Representation of Dynamical Systems for Reservoir Computing
    A general representation of dynamical systems for reservoir computing Sidney Pontes-Filho∗,y,x, Anis Yazidi∗, Jianhua Zhang∗, Hugo Hammer∗, Gustavo B. M. Mello∗, Ioanna Sandvigz, Gunnar Tuftey and Stefano Nichele∗ ∗Department of Computer Science, Oslo Metropolitan University, Oslo, Norway yDepartment of Computer Science, Norwegian University of Science and Technology, Trondheim, Norway zDepartment of Neuromedicine and Movement Science, Norwegian University of Science and Technology, Trondheim, Norway Email: [email protected] Abstract—Dynamical systems are capable of performing com- TABLE I putation in a reservoir computing paradigm. This paper presents EXAMPLES OF DYNAMICAL SYSTEMS. a general representation of these systems as an artificial neural network (ANN). Initially, we implement the simplest dynamical Dynamical system State Time Connectivity system, a cellular automaton. The mathematical fundamentals be- Cellular automata Discrete Discrete Regular hind an ANN are maintained, but the weights of the connections Coupled map lattice Continuous Discrete Regular and the activation function are adjusted to work as an update Random Boolean network Discrete Discrete Random rule in the context of cellular automata. The advantages of such Echo state network Continuous Discrete Random implementation are its usage on specialized and optimized deep Liquid state machine Discrete Continuous Random learning libraries, the capabilities to generalize it to other types of networks and the possibility to evolve cellular automata and other dynamical systems in terms of connectivity, update and learning rules. Our implementation of cellular automata constitutes an reservoirs [6], [7]. Other systems can also exhibit the same initial step towards a general framework for dynamical systems. dynamics. The coupled map lattice [8] is very similar to It aims to evolve such systems to optimize their usage in reservoir CA, the only exception is that the coupled map lattice has computing and to model physical computing substrates.
    [Show full text]
  • EMERGENT COMPUTATION in CELLULAR NEURRL NETWORKS Radu Dogaru
    IENTIFIC SERIES ON Series A Vol. 43 EAR SCIENC Editor: Leon 0. Chua HI EMERGENT COMPUTATION IN CELLULAR NEURRL NETWORKS Radu Dogaru World Scientific EMERGENT COMPUTATION IN CELLULAR NEURAL NETWORKS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Volume 27: The Thermomechanics of Nonlinear Irreversible Behaviors — An Introduction G. A. Maugin Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wiercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L O. Chua Volume 32: From Order to Chaos II L. P. Kadanoff Volume 33: Lectures in Synergetics V. I. Sugakov Volume 34: Introduction to Nonlinear Dynamics* L Kocarev & M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L. Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda Volume 37: Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & O. V. Malysheva Volume 38: Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L Fortuna & D. Porto Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda Volume 40: Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin Volume 41: Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu Volume 42: Chaotic Synchronization: Applications to Living Systems £.
    [Show full text]
  • Writing the History of Dynamical Systems and Chaos
    Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as
    [Show full text]
  • Edge of Spatiotemporal Chaos in Cellular Nonlinear Networks
    1998 Fifth IEEE International Workshop on Cellular Neural Networks and their Applications, London, England, 14-17 April, 1998 EDGE OF SPATIOTEMPORAL CHAOS IN CELLULAR NONLINEAR NETWORKS Alexander S. Dmitriev and Yuri V. Andreyev Institute of Radioengineering and Electronics of the Russian Academy of Sciences Mokhovaya st., 11, Moscow, GSP-3, 103907, Russia Email: [email protected] ABSTRACT: In this report, we investigate phenomena on the edge of spatially uniform chaotic mode and spatial temporal chaos in a lattice of chaotic 1-D maps with only local connections. It is shown that in autonomous lattice with local connections, spatially uni- form chaotic mode cannot exist if the Lyapunov exponent l of the isolated chaotic map is greater than some critical value lcr > 0. We proposed a model of a lattice with a pace- maker and found a spatially uniform mode synchronous with the pacemaker, as well as a spatially uniform mode different from the pacemaker mode. 1. Introduction A number of arguments indicates that along with the three well-studied types of motion of dynamic systems — stationary state, stable periodic and quasi-periodic oscillations, and chaos — there is a special type of the system behavior, i.e., the behavior at the boundary between the regular motion and chaos. As was noticed, pro- cesses similar to evolution or information processing can take place at this boundary, as is often called, on the "edge of chaos". Here are some remarkable examples of the systems demonstrating the behavior on the edge of chaos: cellu- lar automata with corresponding rules [1], self-organized criticality [2], the Earth as a natural system with the main processes taking place on the boundary between the globe and the space.
    [Show full text]
  • The Edge of Chaos – an Alternative to the Random Walk Hypothesis
    THE EDGE OF CHAOS – AN ALTERNATIVE TO THE RANDOM WALK HYPOTHESIS CIARÁN DORNAN O‟FATHAIGH Junior Sophister The Random Walk Hypothesis claims that stock price movements are random and cannot be predicted from past events. A random system may be unpredictable but an unpredictable system need not be random. The alternative is that it could be described by chaos theory and although seem random, not actually be so. Chaos theory can describe the overall order of a non-linear system; it is not about the absence of order but the search for it. In this essay Ciarán Doran O’Fathaigh explains these ideas in greater detail and also looks at empirical work that has concluded in a rejection of the Random Walk Hypothesis. Introduction This essay intends to put forward an alternative to the Random Walk Hypothesis. This alternative will be that systems that appear to be random are in fact chaotic. Firstly, both the Random Walk Hypothesis and Chaos Theory will be outlined. Following that, some empirical cases will be examined where Chaos Theory is applicable, including real exchange rates with the US Dollar, a chaotic attractor for the S&P 500 and the returns on T-bills. The Random Walk Hypothesis The Random Walk Hypothesis states that stock market prices evolve according to a random walk and that endeavours to predict future movements will be fruitless. There is both a narrow version and a broad version of the Random Walk Hypothesis. Narrow Version: The narrow version of the Random Walk Hypothesis asserts that the movements of a stock or the market as a whole cannot be predicted from past behaviour (Wallich, 1968).
    [Show full text]
  • Transformations)
    TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof.
    [Show full text]
  • Control of Chaos: Methods and Applications
    Automation and Remote Control, Vol. 64, No. 5, 2003, pp. 673{713. Translated from Avtomatika i Telemekhanika, No. 5, 2003, pp. 3{45. Original Russian Text Copyright c 2003 by Andrievskii, Fradkov. REVIEWS Control of Chaos: Methods and Applications. I. Methods1 B. R. Andrievskii and A. L. Fradkov Institute of Mechanical Engineering Problems, Russian Academy of Sciences, St. Petersburg, Russia Received October 15, 2002 Abstract|The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system ex- citation, the method of Poincar´e map linearization (OGY method), and the method of time- delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications. 1. INTRODUCTION The term control of chaos is used mostly to denote the area of studies lying at the interfaces between the control theory and the theory of dynamic systems studying the methods of control of deterministic systems with nonregular, chaotic behavior. In the ancient mythology and philosophy, the word \χαωσ" (chaos) meant the disordered state of unformed matter supposed to have existed before the ordered universe. The combination \control of chaos" assumes a paradoxical sense arousing additional interest in the subject.
    [Show full text]
  • Mitra S, Kulkarni S, Stanfield J. Learning at the Edge of Chaos: Self-Organising Systems in Education
    Mitra S, Kulkarni S, Stanfield J. Learning at the Edge of Chaos: Self-Organising Systems in Education. In: Lees, HE; Noddings, N, ed. The Palgrave International Handbook of Alternative Education. London, UK: Palgrave Macmillan, 2016, pp.227-239. Copyright: This extract is taken from the author's original manuscript and has not been edited. The definitive, published, version of record is available here: http://www.palgrave.com/us/book/9781137412904 for product on www.palgrave.com and www.palgraveconnect.com. Please be aware that if third party material (e.g. extracts, figures, tables from other sources) forms part of the material you wish to archive you will need additional clearance from the appropriate rights holders. Date deposited: 13/10/2016 Embargo release date: 19 September 2019 Newcastle University ePrints - eprint.ncl.ac.uk Learning at the edge of chaos - self-organising systems in education Abstract: This chapter takes stock of the evolution of the current primary education model and the potential afforded by current technology and its effects on children. It reports the results of experiments with self-organising systems in primary education and introduces the concept of a Self- Organised Learning Environment (SOLEs). It then describes how SOLEs operate and discusses the implications of the physics of complex systems and their possible connection with self-organised learning amongst children. The implications for certification and qualifications in an internet- immersive world are also discussed. Key words: SOLE, self-organised learning environments, edge of chaos Authors: Sugata Mitra is Professor of Educational Technology and Director of SOLE Central at Newcastle University.
    [Show full text]
  • Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations
    Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations Melanie Mitchell1, Peter T. Hraber1, and James P. Crutchfield2 In Complex Systems, 7:89-130, 1993 Abstract We present results from an experiment similar to one performed by Packard [24], in which a genetic algorithm is used to evolve cellular automata (CA) to perform a particular computational task. Packard examined the frequency of evolved CA rules as a function of Langton's λ parameter [17], and interpreted the results of his experiment as giving evidence for the following two hypotheses: (1) CA rules able to perform complex computations are most likely to be found near \critical" λ values, which have been claimed to correlate with a phase transition between ordered and chaotic behavioral regimes for CA; (2) When CA rules are evolved to perform a complex computation, evolution will tend to select rules with λ values close to the critical values. Our experiment produced very different results, and we suggest that the interpretation of the original results is not correct. We also review and discuss issues related to λ, dynamical-behavior classes, and computation in CA. The main constructive results of our study are identifying the emergence and competition of computational strategies and analyzing the central role of symmetries in an evolutionary system. In particular, we demonstrate how symmetry breaking can impede the evolution toward higher computational capability. 1Santa Fe Institute, 1660 Old Pecos Trail, Suite A, Santa Fe, New Mexico, U.S.A. 87501. Email: [email protected], [email protected] 2Physics Department, University of California, Berkeley, CA, U.S.A.
    [Show full text]
  • Nonlinear Dynamics and Chaos
    1 Nonlinear Dynamics and Chaos: Applications in Atmospheric Sciences A.M.Selvam1 Deputy Director (Retired) Indian Institute of Tropical Meteorology, Pune 411005, India Email: [email protected] Web sites: http://www.geocities.ws/amselvam http://amselvam.tripod.com/index.html http://amselvam.webs.com Abstract Atmospheric flows, an example of turbulent fluid flows, exhibit fractal fluctuations of all space-time scales ranging from turbulence scale of mm - sec to climate scales of thousands of kilometers – years and may be visualized as a nested continuum of weather cycles or periodicities, the smaller cycles existing as intrinsic fine structure of the larger cycles. The power spectra of fractal fluctuations exhibit inverse power law form signifying long - range correlations identified as self - organized criticality and are ubiquitous to dynamical systems in nature and is manifested as sensitive dependence on initial condition or ‘deterministic chaos’ in finite precision computer realizations of nonlinear mathematical models of real world dynamical systems such as atmospheric flows. Though the self- similar nature of atmospheric flows have been widely documented and discussed during the last three to four decades, the exact physical mechanism is not yet identified. There now exists an urgent need to develop and incorporate basic physical concepts of nonlinear dynamics and chaos into classical meteorological theory for more realistic simulation and prediction of weather and climate. A review of nonlinear dynamics and chaos in meteorology and atmospheric physics is summarized in this paper. Key words: Nonlinear dynamics and chaos, Weather and climate prediction, Fractals, Self-organized criticality, Long-range correlations, Inverse power law 1 Corresponding author address: (Res.) Dr.Mrs.A.M.Selvam, B1 Aradhana, 42/2A Shivajinagar, Pune 411005, India.
    [Show full text]
  • Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction
    systems Communication Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction Geoff Boeing Department of City and Regional Planning, University of California, Berkeley, CA 94720, USA; [email protected]; Tel.: +1-510-642-6000 Academic Editor: Ockie Bosch Received: 7 September 2016; Accepted: 7 November 2016; Published: 13 November 2016 Abstract: Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior. Keywords: visualization; nonlinear dynamics; chaos; fractal; attractor; bifurcation; dynamical systems; prediction; python; logistic map 1. Introduction Chaos theory is a branch of mathematics that deals with nonlinear dynamical systems.
    [Show full text]
  • ON ORDER and RANDOMNESS: a VIEW from the EDGE of CHAOS Gennady Shkliarevsky Bard College “[QM] Yields Much, but It Hardly Brings Us Closer to the Old One's Secrets
    ON ORDER AND RANDOMNESS: A VIEW FROM THE EDGE OF CHAOS Gennady Shkliarevsky Bard College “[QM] yields much, but it hardly brings us closer to the Old One's secrets. I, in any case, am convinced that He does not play dice.” --A. Einstein “Not only does God definitely play dice, but He sometimes confuses us by throwing them where they can't be seen.” --S. Hawking Introduction Production of knowledge is undoubtedly one of the most defining features of human civilization. We take enormous pride in how much we know and how much we can control the world in which we live. We also greatly depend on the production of knowledge in ensuring our prosperity and wellbeing, safety and security, our very future. Indeed, it is no exaggeration to say that our very sense of dignity and self-worth is due in no small degree to our unique capacity to generate knowledge. There is a close relationship between production of knowledge and critical introspection into the ways we know. Our mind does not have direct access to reality and all our knowledge is mediated. The fact that we understand reality through the prism of our mind and its constructs implies that what we know critically depends on the tools we use, including our mental tools. Every advance in our knowledge has always involved critical rethinking of the way we know. What we know about how we know and how well we control our mental tools has a decisive effect on our knowledge production. Without such critical introspection our knowledge can stagnate and become obsolete.
    [Show full text]